9801/9800: Difference between revisions
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'''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[11-limit]] [[unnoticeable comma]] measuring about 0.18{{cent}}. It is the smallest 11-limit [[superparticular]] interval. | '''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[11-limit]] [[unnoticeable comma]] measuring about 0.18{{cent}}. It is the smallest 11-limit [[superparticular]] interval. | ||
It can be described as the difference between [[99/98]] and [[100/99]], and between [[99/70]] and its octave complement, [[140/99]]. It is also the difference between [[245/243]] and [[121/120]], and a stack of two 11/7's and 81/80 against 5/2 | It can be described as the difference between [[99/98]] and [[100/99]], and between [[99/70]] and its octave complement, [[140/99]]. It is also the difference between [[245/243]] and [[121/120]], and a stack of two 11/7's and 81/80 against 5/2. | ||
It factors into the two smallest 13-limit superparticular commas: 9801/9800 = 10648/10647 × 123201/123200. | It factors into the two smallest 13-limit superparticular commas: 9801/9800 = 10648/10647 × 123201/123200. | ||
== Temperaments == | == Temperaments == | ||
Tempering | [[Tempering out]] this comma leads to the '''kalismic temperament''', which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Tempering it out also means that [[10/9]] and [[11/7]] are 1/2-octave apart, as well as are [[11/10]] and [[14/9]]. Odd-numbered edos cannot temper it out. See [[Rank-4 temperament #Kalismic (9801/9800)]] for some technical details. See [[Kalismic temperaments]] for a collection of rank-3 temperaments where it is tempered out. | ||
== Etymology == | == Etymology == | ||
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== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
Revision as of 11:19, 5 November 2024
| Interval information |
reduced
S33 / S35
9801/9800, the kalisma, sometimes described as Gauss' comma, is an 11-limit unnoticeable comma measuring about 0.18 ¢. It is the smallest 11-limit superparticular interval.
It can be described as the difference between 99/98 and 100/99, and between 99/70 and its octave complement, 140/99. It is also the difference between 245/243 and 121/120, and a stack of two 11/7's and 81/80 against 5/2.
It factors into the two smallest 13-limit superparticular commas: 9801/9800 = 10648/10647 × 123201/123200.
Temperaments
Tempering out this comma leads to the kalismic temperament, which splits the octave into two equal parts, each representing 99/70~140/99. Tempering it out also means that 10/9 and 11/7 are 1/2-octave apart, as well as are 11/10 and 14/9. Odd-numbered edos cannot temper it out. See Rank-4 temperament #Kalismic (9801/9800) for some technical details. See Kalismic temperaments for a collection of rank-3 temperaments where it is tempered out.
Etymology
This comma was named kalisma by Margo Schulter in 2000 from the Greek root kal- ("beautiful")[1]. Gene Ward Smith, not aware of Margo's work, proposed gaussisma in 2004, reasoning that D. H. Lehmer claimed Carl Friedrich Gauss had mentioned the ratio[2].